Análisis de los resultados

No structure was imposed so far on the analogy categorization. Understanding how individ- uals categorize contingencies to form their expectations is clearly beyond the scope of this paper (it is at the heart of a large body of the ongoing research of cognitive scientists, see Holyoak-Thagard 1995 and Dunbar 2000, for example). As a modest game-theoretic inves- tigation, we now review two principles (for analogy partitionings) that may alternatively be viewed as attempts to reÞne the concept of analogy-based expectation equilibrium.

Analogy expectation and similar play:

An appealing idea seems to be that in order for player ito pool several nodes (j, h) into a single class of analogy, player i should himself consider playing in the same way in some pool of nodes. One difficulty is that in general player ineed not move in the same nodes as player j, and therefore one should also worry about which nodesh0 _∈ Hi player i considers

as being similar to nodesh_∈Hj.

A class of situations in which this issue can be addressed simply is one in which whenever player ibundles two elements (j, h) and (j0, h0) into the same analogy class αi, player i also

has to move inhand h0_{. And playeriplays the same behavioral strategy at nodeshand h}0_. We distinguish according to whether this property should be met only for histories reached along the played path or for all histories.55

In the Þnitely repeated prisoner’s dilemma P DT in which player 1 uses two classes of

analogyα1,α01, (according to whether or not at least oneDwas played earlier) and player 2 uses theÞnest partition, we observed (see Remark 1 after Proposition 7) that for all histories

hmet along the played path, whenever (2, h)_∈α1 (resp. α0₁), player 1 plays the same action

C (resp. D) at nodeh. Thus, the property is met for all historieshreachedalong the played path.56

The next example is such that the property is met for all histories (whether on or offthe equilibrium path), and yet the play differs from that of the Subgame Perfect Nash Equilib- rium:

Example 3: Consider the following two-stage two-player game. Player 1 movesÞrst and chooses between the normal form game G or G0. In both G and G0, players 1 and 2 move simultaneously, and in both Gand G0, player 1 chooses in A1 ={U, D}, player 2 chooses in

{L, R_}. We assume that U is a dominant strategy in both G and G0 for player 1. Player 2’s best-response toU is Rin gameG, whereas it isL in game G0. Finally, we assume that

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One might argue that a player is more likely to have doubts about his analogy partitioning if the property is violated on the equilibrium path histories.

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The requirement is not met though for the offthe equilibrium path historyh= (a(t))_tT₌₁−1witha(t)= (C, C) for allt, in which playeriwould playD(and notC as for the other histories inα1).

player 1 derives a higher payoffwhen (U, R) is played in game Gthan when (U, L) is played in gameG0. And that player 1 derives a higher payoffwhen (U, L) is played in gameG0 than when (U, L) is played in game G.

The unique Subgame Perfect Nash Equilibrium is such that player 1 chooses gameGand then (U, R) occurs (this yields player 1 more than (U, L) inG0).

Suppose that player 1 puts in the same analogy class (2, G) and (2, G0) in order to predict player 2’s behavior. NoteÞrst that player 1 behaves in the same way inGandG0 (he has the same dominant strategy in both games). Thus, the required property is satisÞed. Second, it is readily veriÞed that an equilibrium outcome in this analogy setting is that player 1 chooses

G0 (expecting player 2 to playL in both Gand G0), since player 1 prefers (U, L) in game G0

rather than (U, L) in game G.

All analogy classes must be reached:

Another property that may be of interest is that players structure their analogy classes so that each analogy class is reached with positive probability in equilibrium.57 The next Proposition provides some insight about the effect of this principle in the centipede game

CPK considered in subsection 3.2.

Proposition 11 Let (σ,β) be an analogy-based expectation equilibrium of (N, CPK,%i, An)

whereN =_{1,2_}denotes the set of players,%iplayer i’s preferences, andAnthe partitioning

proÞle used by the players. Suppose that for all k, 1₂ak−2+ 1₂ak−1 > ak and 1₂bk−2+ 1₂bk−1 >

bk. If σ employs only pure strategies, andallanalogy classes of both players are reached with

positive probability according to σ, then the equilibrium outcome is that player 1 Takes in the last node N₁(1).

Proof. Take at node N₁(1) is a possible equilibrium outcome when players use the coarsest partition (see subsection 3.2). Since all classes of both players are then reached with positive probability, this outcome can be sustained in the way required by the Proposition.

Suppose that another outcome, i.e. player iTakes at nodeN_i(k) with (i, k)₆= (1,1), were to emerge with the same requirements.

First, it cannot be that this outcome corresponds to the Subgame Perfect Nash Equi- librium outcome, since then no node N₁(k) would be reached, and thus at least one of the analogy classes of player 2 would not be reached in equilibrium.

If playeri were to Pass at nodeN_i(k) this would lead to nodeN_j(k0),j ₆=i, withk0 =kif

i= 1 and k0 _{=k−1 if i = 2. Since node N}(k0)

j is not reached in equilibrium and since all

57_{A possible psychological rationale for this is that players tend to prefer structuring analogy classes so that}

analogy classes must be reached with positive probability, it must be that there is an analogy class αi of player isuch that (j, N(k

0₎ j )∈αi and (j, N (k00₎ j )∈αi wherek00 < k0 (nodes N (k00₎ j

withk00 > k0 are not reached).58 Since at any node N_j(k00) withk00< k0 player j Passes with

probability 1 (remember that Take at node N_i(k) is the assumed outcome), it must be that the analogy-based expectation of player isatisÞes

βi(αi) =λi·P ass+ (1−λi)·T akewithλi ≥

1 2.

But given this expectation (and given that for allk, 1₂ak−2+ 1₂ak−1 > akand 1₂bk−2+ 1₂bk−1 >

bk), Taking at nodeNi(k)cannot be a best-response toβi(at nodeNi(k), playerishould strictly

prefer Passing rather than Taking). This leads to a contradiction.

4.3

Multiplicity and Analogy-Based Expectation Equilibrium

In this subsection we would like to highlight the implication of analogy reasoning on the multiplicity of equilibria. A Þrst observation is that the analogy treatment may sometimes kill the multiplicity of equilibria that would otherwise prevail. For example, in the inÞnitely repeated prisoner’s dilemmaP Dδ, we observed that if one player has an analogy partitioning such that his own actions play no role, then the only equilibrium outcome is the repetition of (D, D). Here, the analogy treatment kills the multiplicity because it does not permit enough conditioning of players’ expectations.

A second observation is that the analogy treatment may sometimes create a multiplicity of equilibria by permitting some form of conditioning that would not be possible otherwise. For example, in theÞnitely repeated prisoner’s dilemma in which player 1 uses two classes according to whether or not at least one D was played earlier and player 2 uses the Þnest partitioning, we saw an equilibrium in which both players playCexcept in the last two periods (see subsection 3.3.). But, there is also an equilibrium for this partitioning in which both players playD in every period. Here, the multiplicity arises because the analogy treatment permits a conditioning of player 1’s expectation (upon whether or not at least one D was played earlier) that would not be possible otherwise (due to the logic of backward induction). A third observation is that the consistency condition (1) implies non-linearities, thus creating a potential for multiple equilibria, even when players use a single class of analogy59 and there is a unique equilibrium in the setup without analogy. For example, in the centipede game CPK in which both players use the coarsest partition and condition (4) holds, we saw

that there are two pure strategy analogy-based expectation equilibria (see Proposition 3).

58_{There exists at least one such node because (i, k)}

6

= (1,1).

59